This paper introduces new solvers for efficiently computing solutions to large-scale inverse problems with group sparsity regularization, including both non-overlapping and overlapping groups. Group sparsity regularization refers to a type of structured sparsity regularization, where the goal is to impose additional structure in the regularization process by assigning variables to predefined groups that may represent graph or network structures. Special cases of group sparsity regularization include $\ell_1$ and isotropic total variation regularization. In this work, we develop hybrid projection methods based on flexible Krylov subspaces, where we first recast the group sparsity regularization term as a sequence of 2-norm penalization terms using adaptive regularization matrices in an iterative reweighted norm fashion. Then we exploit flexible preconditioning techniques to efficiently incorporate the weight updates. The main advantages of these methods are that they are computationally efficient (leveraging the advantages of flexible methods), they are general (and therefore very easily adaptable to new regularization term choices), and they are able to select the regularization parameters automatically and adaptively (exploiting the advantages of hybrid methods). Extensions to multiple regularization terms and solution decomposition frameworks (e.g., for anomaly detection) are described, and a variety of numerical examples demonstrate both the efficiency and accuracy of the proposed approaches compared to existing solvers.

翻译：本文介绍了用于高效计算大规模反问题中组稀疏正则化（包括非重叠组和重叠组）的新求解器。组稀疏正则化是一种结构化稀疏正则化，其目标是通过将变量分配到预定义的组（可能表示图或网络结构）来在正则化过程中施加额外结构。组稀疏正则化的特例包括$\ell_1$范数和各向同性全变分正则化。本研究开发了基于灵活Krylov子空间的混合投影方法：首先，利用自适应正则化矩阵以迭代重加权范数方式将组稀疏正则化项转化为一系列2-范数惩罚项；随后，采用灵活预处理技术高效地纳入权重更新。这些方法的主要优势在于：计算效率高（利用灵活方法的优势）、通用性强（易于适配新的正则化项选择），以及能够自动自适应地选择正则化参数（利用混合方法的优势）。本文还描述了向多正则化项和求解分解框架（如异常检测）的扩展，并通过多种数值算例验证了所提方法相比现有求解器的效率与精度。